Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies ω$$1$$ and ω$$2$$ and have total energies $$E1$$ and $$E2$$ ,respectively. The variations of their momenta P with positions x are shown in the figures. If $$ab=n2$$ and $$aR=n$$, then the correct $$equation(s)$$ is (are)

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Two independent  harmonic oscillators of equal mass are oscillating about the origin with angular frequencies ω$$1$$  and  ω$$2$$ and have total energies $$E1$$  and $$E2$$ ,respectively. The variations of their momenta P  with positions  x are shown in the figures. If $$ab=n2$$ and  $$aR=n$$, then the correct $$equation(s)$$ is (are)

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$$1st$$ Particle:  P $$=$$ $$0$$ at x $$=$$ a ⇒ ‘a’ is the amplitude of Oscillation '$$A1$$' .  At x $$=$$ $$0$$ , P $$=$$ b (at$$ mean $$position) ⇒$$mvmax=b$$ .  ⇒$$vmax=bm----(1)$$ $$E1=12mvmax2=m2bm2=b22m$$∵$$vmax=bm$$ Also, $$vmax=A1$$ω$$1=bm$$​⇒ω$$1=bma=1mn2$$∵$$a=bn2$$−−−−−$$2$$ $$2nd$$ Particle:P $$=$$ $$0$$ at x $$=$$ R  ⇒$$A2=RAt$$ x $$=$$ $$0$$,  $$P=Rmvmax=R$$​⇒$$vmax=Rm$$−−−−$$3E2=12mvmax2=m2Rm2=R22m$$∵$$vmax=Rm----(4)$$  Also, $$vmax=A2$$ω$$2=Rm$$​⇒ω$$2=RmR=1m$$−−−−$$5$$ (A) $$E1$$ω$$1=b22m$$×$$1mn2$$​$$E2$$ω$$2=R22m$$×$$1m$$​We can see, $$E1$$ω$$1$$≠$$E2$$ω$$2(B)$$  ω$$2$$ω$$1=1/m1/mn2=n2(C)$$  ω$$1$$ω$$2=1mn2$$×$$1m=1m2n2C$$ is not $$correct(D)$$ $$E1$$ω$$1=b2/2m1/mn2=b2n22=a22n2=R22$$∵$$a=bn2$$ & $$aR=n$$  $$E2$$ω$$2=R2/2m1/m=R22$$⇒$$E1$$ω$$1=E2$$ω$$2$$  (D) is correct

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